Abstract: In this series of lectures, we will discuss basic examples of compact quantum groups and their (finite dimensional) representations. We will start with reviewing the classical theory. We shall classify all finite dimensional irreducible representations of compact Lie groups SU(2) and SU(3). Then we will proceed to the general theory of representation of compact Lie groups and will discuss several important results including the highest weight theory, the Peter-Weyl decomposition theorem, and also the Borel-Weil-Bott construction of representations. Finally, we will see how much of the theory holds in the quantum case.

Abstract: In many geometric situations we may encounter the action of a group G on a manifold M, e.g., in the context of foliations. If the action is free and proper, then the quotient M/G is a smooth manifold. However, in general the quotient M/G need not even be Hausdorff. Under these conditions how can we do diffeomorphism-invariant geometry?

Noncommutative geometry provides us with a solution by trading the badly behaved space M/G for a non-commutative algebra, which essentially plays the role of the algebra of smooth functions on that space. The local index formula of Atiyah and Singer ultimately holds in the setting of noncommutative geometry. This enabled Connes and Moscovici to reformulation of the local index formula in the setting of diffeomorphism-invariant geometry.

The first part of the lectures will be a review of noncommutative geometry and Connes-Moscovici's index theorem in diffeomorphism-invariant geometry.

In the 2nd part, I will hint to on-going projects on the reformulation of the local index formula in two new geometric settings: biholomorphism-invariant geometry of strictly pseudo-convex domains and contactomorphism-invariant geometry of contact manifolds.

Abstract: A Gerstenhaber algebra is a special kind of graded Poisson algebra. A homotopy Gerstenhaber algebra is a specific "up to homotopy" version of the former. Important examples of homotopy Gerstenhaber algebras are the Hochschild cochains of an associative algebra and the cochain complex of a simplicial set.

In this talk I will address the following problem: What structure does the tensor product of two homotopy Gerstenhaber algebras have? If time permits, I will also talk about formality results for homotopy Gerstenhaber algebras.